2002 Fiscal Year Final Research Report Summary
Geometry on String Theory and Moduli spaces
| Project/Area Number |
12440008
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kobe University |
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Principal Investigator |
SAITO Masa-hiko Kobe University Faculty of Science Professor, 理学部, 教授 (80183044)
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| Co-Investigator(Kenkyū-buntansha) |
HOSONO Shinobu Tokyo University Grad. Sch. Of Math. Science Associate Professor, 数理科学研究科, 助教授 (60212198)
YAMADA Yasuhiko Kobe University Faculty of Science Professor, 理学部, 教授 (00202383)
NOUMI Masatoshi Kobe University Grad. Sch. Of Sci. and Tech. Professor, 自然科学研究科, 教授 (80164672)
YOSHIOKA Kota Kobe University Faculty of Science Associate Professor, 理学部, 助教授 (40274047)
MUKAI Shigeru Kyoto University RIMS Professor, 数理解析研究所, 教授 (80115641)
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| Project Period (FY) |
2000 – 2002
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| Keywords | BPS invariants / Gopakumar-Vafa conjecture / Homological Mirror Symmetry / Calabi-Yau manifolds / Gromov-Witten invariants / Okamoto-Painleve pairs / Painleve equations / Moduli spaces of Vector Bundles |
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| Research Abstract |
During the period of the project, we have investigated the following subjects and obtained the following results. (i) Gromov-Witten invariants and BPS invariants for Calabi-Yau manifolds-Gopakumar-Vafa conjecture, (ii) Homological Mirror Symmetry and Geometry of derived categories, (iii) Algebraic Geometry of spaces of initial conditions of Painleve equations, (iv) Lie theoretic studies for Painleve equations and its generalizations, (v) Symmetries of the moduli spaces of vector bundles, (vi) New development of invariant theory. As for (i), we proposed a mathematical definition of BPS invariants for Calabi-Yau 3-fold and verified their compatibility for the known calculation of Gromov-Witten invariants assuming Gopakumar-Vafa conjecture. In (iii), we introduce the notion of Okamoto-Painleve pairs and proved that one can derive Painleve equations through deformation theory of Okamoto-Painleve pairs. As for the studies of Painleve equations (iv), our group have been developing Lie theoretic approach and algebro-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.
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